M×Φ

Annals of Mathematics and Philosophy

Volume III, number 1, 2025

Volume III, number 1, 2025
Cite this issue
M×Φ — Annals of Mathematics and Philosophy, 3(1), 2025.

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Articles

Who Proved the Independence of the Continuum Hypothesis?

Benjamin Callard, Palle Yourgrau pp. 1–21

Paul Cohen is routinely represented as having proved the independence of the Continuum Hypothesis from the axioms of Zermelo-Fraenkel set theory — despite the equally uncontroversial, and apparently contradictory, concession that Cohen proved only one of the two conditions on independence, Kurt Gödel having proved the other. In this essay we explore, and argue for a position on, this strange and unsatisfactory situation, and suggest that our position generalizes in ways that would upset the current conventions governing the assignment of credit for intellectual discoveries.

Callard, B., & Yourgrau, P. (2025). Who Proved the Independence of the Continuum Hypothesis?. M×Φ — Annals of Mathematics and Philosophy, 3(1), 1–21.

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Explanations of Mathematical Statements

Joseph Y. Halpern pp. 23–42

A definition of what counts as an explanation of a mathematical statement, and when one explanation is better than another, is given. Since all mathematical facts must be true in all causal models, and hence known by an agent, mathematical facts cannot be part of an explanation (under the standard notion of explanation). This problem is solved using impossible possible worlds.

Halpern, J. (2025). Explanations of Mathematical Statements. M×Φ — Annals of Mathematics and Philosophy, 3(1), 23–42.

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The philosophical significance of algebraic geometry

Alberto Peruzzi pp. 43–82

This paper explores the philosophical significance of algebraic geometry by addressing Federigo Enriques' question on the relation between logic and intuition. Through a historical and conceptual analysis, it traces the transition from the Italian school of algebraic geometry to the abstract frameworks developed by Grothendieck and Lawvere. The article highlights how key categorical notions — such as schemes, sheaves, and toposes — transform the interplay between geometry and logic, allowing logical principles to be internalized within geometric structures. It argues that the philosophy of mathematics cannot be reduced to meta-mathematical reflection alone, since algebraic geometry itself generates conceptual innovations with direct philosophical import. Ultimately, the paper shows that algebraic geometry reshapes the foundations of mathematics by dissolving the separation between formal rigor and spatial intuition and providing support to the reasonable effectiveness of "conceptual mathematics". This approach not only provides an answer to Enriques' question but also defines a new sense for the foundations of mathematics, where logical principles are intrinsically linked to the geometric structure of a mathematical universe.

Peruzzi, A. (2025). The philosophical significance of algebraic geometry. M×Φ — Annals of Mathematics and Philosophy, 3(1), 43–82.

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Human mathematics in the age of reasoning machines

Akshay Venkatesh pp. 83–116

This essay intertwines reflection on current mathematical practice with discussion of how it may be reshaped by automation. I speculate that the conceptual language of mathematics might undergo drastic rewritings, and look to historical examples for guidance.

Venkatesh, A. (2025). Human mathematics in the age of reasoning machines. M×Φ — Annals of Mathematics and Philosophy, 3(1), 83–116.

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Reeditions and translations

Introduction to the English translation of « Il significato della critica dei principi »

Frédéric Patras pp. 117–118

In this article, Enriques focuses on the critique of principles and their role in the development of mathematics, a role whose analysis and understanding are inseparable from the adoption of a historical perspective. The questioning and reworking of fundamental concepts emerge as one of the essential components of mathematical progress, providing it with ever more refined and profound tools.

Patras, F. (2025). Introduction to the English translation of « Il significato della critica dei principi ». M×Φ — Annals of Mathematics and Philosophy, 3(1), 117–118.

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The significance of criticism of principles in the development of mathematics

Federigo Enriques pp. 119–138

First English translation of Enriques’s classic 1912 essay on the critical examination of mathematical principles as a driving force in the development of mathematics.

Enriques, F. (2025). The significance of criticism of principles in the development of mathematics. M×Φ — Annals of Mathematics and Philosophy, 3(1), 119–138.

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